Solution: To find when the gears align again, we compute the least common multiple (LCM) of their rotation periods. Since they rotate at 48 and 72 rpm (rotations per minute), the time until alignment is the time it takes for each to complete a whole number of rotations such that both return to start simultaneously. This is equivalent to the LCM of the number of rotations per minute in terms of cycle time. First, find the LCM of the rotation counts over time or convert to cycle periods: The time for one rotation is $ \frac{1}{48} $ minutes and $ \frac{1}{72} $ minutes. So we find $ \mathrm{LCM}\left(\frac{1}{48}, \frac{1}{72}\right) = \frac{1}{\mathrm{GCD}(48, 72)} $. Compute $ \mathrm{GCD}(48, 72) $:

Sat, 02 May 2026 20:14:54 +0000

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